Observable Sets, Potentials and Schrödinger Equations

Consider the Schrödinger equation: i ∂ t u = H u over R n , where H is a self-adjoint operator on L 2 ( R n ) which is the sum of - Δ and some potential. This paper aims to study the observability for the above equation, including observable sets and the observable time. We mention that a measurable subset E ⊂ R n is called an observable set at time T > 0 for the above equation, if there is a constant C > 0 (depending on T and E ) such that ∫ R n | u 0 ( x ) | 2 d x ≤ C ∫ 0 T ∫ E | e - i t H u 0 | 2 d x d t for all u 0 ∈ L 2 ( R n ) . First, we characterize observable sets for the 1-dim case where H = - ∂ x 2 + x 2 m (with m ∈ N : = { 0 , 1 , ⋯ } ). More precisely, we obtain what follows: ( i ) When m = 0 , E ⊂ R is an observable set at some time if and only if it is thick, namely, there are constants γ , L > 0 so that E ⋂ [ x , x + L ] ≥ γ L for each x ∈ R ; ( ii ) When m = 1 ( m ≥ 2 resp.), E is an observable set at some time (at any time resp.) if and only if it is weakly thick, namely lim ̲ x → + ∞ | E ⋂ [ - x , x ] | x > 0 . These reveal how potentials x 2 m affect the observability. Second, we obtain what follows for the n -dim case where H = - Δ + | x | 2 (the Harmonic oscillator): ( i ) For each r > 0 , the exterior domain B c ( 0 , r ) is an observable set at any time; ( ii ) Let E 1 be a half of B c ( 0 , r ) bisected by a hyperplane across the origin. Then E 1 is an observable set at time T > 0 if and only if T > π 2 .